Ok. (I actually got this from a friend and it made a lot of sense.)
Before the fall of Adam there was no death, so the dinosaurs had to either exist with Adam, or after him. I think they existed much later, even after the flood. During, and for a while after the flood, the earth was in geologic turmoil, with things happening much faster than usual. Like when we read about the continents moving, or at Christ's death, which are other examples of periods of geologic turmoil.
So did Noah then have the dinosaurs on his arc? Yes, but they were obviously not the size as we think of them. (I do have to sneak evolution in here a bit.) Evolution does exist within species. Humans today are larger than they were a few centuries ago. If conditions are favorable for a particular species, it will thrive and evolve larger. So possibly the conditions on the earth, immediately after the flood, favored the original dinosaur species so well that it thrived and rather quickly evolved to be quite huge.
But the earth was still in its rapid change-mode, conditions continued to change and eventually the dinosaurs died out. With such a large earth and so few people here (because of the flood) they could have well stayed out of each other's way.
That's a brief version. I find it interesting when religion doesn't necessarily contradict science. But this is just a theory.